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Let us now recall the definition of a generic random variable, and then the specific case of discrete random variables.

EXAMPLE 1.12.– Let us return to the experiment where a six-sided die is rolled, where the set of possible outcomes is Ω = {1, 2, 3, 4, 5, 6}, which is endowed with the uniform probability. Consider the following game:

 – if the result is even, you win 10 ;

 – if the result is odd, you win 20 .

This game can be modeled using the random variable defined by:

X

The distribution of X is a probability distribution on (E, ε); it is also called the image distribution of ℙ by X.

i

1N1N

1N

n

n

If the Bernoulli experiment with probability of success p is repeated N times, independently, then the binomial distribution is the distribution of the random variable containing the number of successes at the end of the N repetitions of the experiment.

and for any k ∈ X(Ω),

If we consider an urn containing N indistinguishable balls, k red balls and N – k white balls, with k ∈ {1, ...N 1}, and if we simultaneously draw n balls, then the random variable X, equal to the number of red balls obtained, follows a hypergeometric distribution with parameters N, n and